ABSTRACT
The previous consideration of flows (Chapters 12, 14, 16, 28, 34, 36, and 38 in volume I) has been mostly restricted to plane, steady, irrotational, and incompressible conditions. The compressible flow in homentropic conditions was considered in connection with the sound speed (Subsection I.14.6.1) and the measurement of velocity using Pitot (Venturi) tubes [Subsection I.14.6.2 (Subsection I.14.7.3)]. The homentropic compressible flow limits the velocity and hence sets a minimum radius (Section 2.1) for the flow due to a source, sink, vortex, or their combination in a spiral flow. Another way to avoid the singular velocity at the center of a potential vortex is to match the outer irrotational flow to a rotational core with zero velocity at the center (Section 2.2), for example, rigidly rotating or some other angular or smooth radial profile of the tangential velocity. Unlike the line vortex, for which vorticity is concentrated at the center, in the case of a nonpotential core, the vortical flow occupies a finite or infinite domain. For the same boundary conditions, an irrotational flow has less kinetic energy than a rotational flow (Section 2.3); the equations of motion in intrinsic coordinates, that is, parallel and tangent to the velocity, show that, in a rotational flow, the stagnation pressure is conserved only along streamlines but varies between streamlines due to the vorticity, and thus, Bernoulli’s equation does not apply; although there is no scalar potential, for an incompressible rotational flow, there is a stream function satisfying a Poisson equation (Section 2.4). The Blasius theorem, specifying the lift and drag forces and pitching moment on a body, can be expressed in terms of the stream function alone (Section 2.5); likewise, the effect of inserting a cylinder in a rotational flow is specified by the second circle theorem, in terms of the stream function alone. The combination allows the consideration of a vortical flow past a cylinder (Section 2.6), showing that the mean flow vorticity can add to or subtract from the lift due to the circulation. Returning to assemblies of line monopoles, it can be shown that their centroid moves uniformly (Section 2.7), even though the monopoles move relative to each other; in particular, two monopoles move relative to each other, but can be in static equilibrium in specific conditions, for example, in a duct with parallel walls (Section 2.7) or behind a cylinder in a stream (Section 2.8). A monopole starting outside an equilibrium position follows a path specified by a path function (Section 2.9) that involves the Green’s or influence function.
